In conventional reamers of this type, the guide member is fastened to the cutter head by means of a setscrew which is received in a recess in the cutter head. In this arrangement, the smoothing effect exerted by the guide members on the wall of the bore is decreased by the accummulation of splinters in the recess which accommodates the head of the setscrew. Since the guide members are relatively thin in a radial direction, the danger also exists that the cross section of the guide member will be sufficiently weakened to allow them to bend in the region of the setscrew. Such bending would of course also result in improper abutment of the guide member on the wall of the bore.